Heaviness in Circle Rotations

TL;DR

This paper investigates the structure and Hausdorff dimension of points in the unit interval that, under irrational rotation, always have at least as many points in the lower half as in the upper half over finite orbit segments, using continued fractions.

Contribution

It introduces an inductive method based on continued fractions to describe this set and analyzes its Hausdorff dimension depending on the rotation parameter.

Findings

The set's Hausdorff dimension varies with alpha.

All possible dimensions can be achieved depending on alpha.

The essential infimum of the dimension over alpha is positive.

Abstract

We are concerned with describing the structure of the set of points in the unit interval which, when subjected to rotation by irrational alpha modulo one, for all finite portions of the orbit contain at least as many points in the bottom half of the interval as in the top half. Specifically, an inductive procedure for describing the set based on the continued fraction expansion of alpha is developed, leading into a discussion of the Hausdorff dimension of this set. Depending on the parameter alpha, all possible dimensions may be achieved, and the essential infimum (with respect to alpha) of this dimension is positive.